Cardinal directions are an integral part of every person’s knowledge of the world around them. Everyone intuitively knows that California is west of New York, and that Florida is south of Alaska. The four cardinal directions can even become attractions in their own right, as evidenced by the southernmost point of the lower 48 states in Key West. However, when we think of directions we tend to limit ourselves to North, South, East, and West. I wanted to take a look at the intermediate directions: Northeast, Northwest, Southeast, and Southwest. Not the easy aspects, either: everyone knows that Maine is the farthest state Northeast. No, I wanted to be able to answer more complicated questions, like if Albuquerque or Des Moines was farther southeast. Below is my quest to answer these questions.
Disclaimer: any math you see below, no matter how basic, might be dead wrong. I’m a travel enthusiast, not a genius.
I’ll begin by looking at the standard measurements for cardinal directions: latitude and longitude. A high latitude means you are north, and (in the western hemisphere) a high longitude means you are west (we’ll ignore the fact that in decimal degree notation, this number would be negative). Based on this, there should be a simple series of equations to determine intermediate directions. By adding together the latitude and longitude of a location, you should be able to determine where on the northwest/southeast scale it lies: a higher number means that it is more northwest, while a lower number means it is more southeast. Similarly, by subtracting the longitude from the latitude, you should be able to determine how far northeast or southwest a location is: a higher number here means it is northeast, while a lower one indicates southwest.
However, there are issues with this simple equation. To demonstrate, let’s use the cities of Virginia Beach, Virginia and Savannah, Georgia.
Anyone with eyes can see that Savannah is further southeast. However, let’s take a look at the math. Savannah’s latitude is 32.071 N, while its longitude is 81.098 W; Virginia Beach has a latitude of 36.851 N and a longitude of 75.978 W. The sum for Savannah is 113.169, and for Virginia Beach is 112.829; this would indicate that Virginia Beach is more southeasterly than Savannah!
The reason for this stems from the fact that Virginia Beach is farther up the coast, and therefore at a more eastern longitude, than Savannah. Since there are more total degrees of longitude in the Western Hemisphere than degrees of latitude (180 versus 90), the longitude of a location mattered more than its latitude when calculating intermediate directions. I didn’t like that, so I set about figuring out a better, much more harebrained, solution.
I figured that what I really wanted was a way to map places within the US in relation to the US itself. Therefore, I had to adjust the latitude and longitude of each location relative to the total latitude and longitude of the US. For the purposes of this exercise, I limited myself to the mainland.
First, I would need to know the dimensions of the USA by finding its extreme cardinal points. These turned out to be the following:
Northernmost: 49.384472 at the Northwest Angle of Minnesota
Southernmost: 24.520833 at Ballast Key, Florida
Easternmost: 66.947028 at Quoddy Head, Maine
Westernmost: 124.771629 near Cape Alava, Washington
This meant that the US had a total of 24.863639 degrees of latitude and 57.824666 degrees of longitude. These numbers will be important for later.
Next, I had to adjust each location’s coordinates based on these numbers. To do this, I subtracted the latitude of the southernmost point from the location’s actual latitude and subtracted the longitude of the easternmost point from the actual longitude. Using Savannah and Virginia Beach as examples, I was left with the following coordinates:
Savannah: 7.550167 N, 14.150972 W
Virginia Beach: 12.330167 N, 9.030972 W
Now that I had these adjusted coordinates, I could find out the percentage North and percentage West of each location. I did this by dividing the adjusted latitude and longitude by the total degrees of latitude and longitude shown two paragraphs above. This meant that, compared to the rest of the USA:
Savannah was 30.6% north, and 24.5% west.
Virginia Beach was 49.6% north, and 15.6% West.
Now, finally, I could add these percentages together. The smaller sum should be more southeasterly.
Virginia Beach: 65.2
See? After all that work, Savannah reclaims its rightful place as the more southeastern city! You could also subtract the two numbers to determine which city is more northeastern/southwestern.
I took this exercise a step further by uploading every county’s center point into an excel document. I then used this series of equations to form a table where you can sort each county by its position according to intermediate directions. Enjoy it here!
Some interesting things I found:
- Even with the adjustment, Santa Barbara is farther southwest than Los Angeles simply because it is so much farther west.
- Brownsville, Texas is farther southwest than San Francisco!
- There are points in Michigan’s upper peninsula that are farther northeast than parts of New York.
- Indianapolis is technically farther southeast than Tulsa, but just barely!
I hope you enjoyed this completely unscientific look at my obsession with directions; feel free to let me know if my calculations are wrong!
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